DE10255593A1 - Method and arrangement for the detection and measurement of the phase of periodic biosignals - Google Patents

Abstract

A method and an arrangement are to be specified with which it is possible to detect and measure a causal phase response in periodic biosignals with a better reliability and higher speed than conventional methods with a reduced computing effort. DOLLAR A A condition observer is set up parallel to the analyzed biological system. The output variables of the biological system and the observer are evaluated with the aid of a Kalman filter and used to determine the phase. DOLLAR A The determined phase of a periodic biosignal is used for functional diagnostic purposes.

Description

The invention relates to a method and an arrangement for real-time capability reliable Detection and measurement of the phase of periodic physiological quantities or Biosignals.

Methods are in the prior art Known to determine the phase one over time of the biosignal sliding or sequentially arranged analysis window use. On the signal section lying in the analysis window methods based on the Fourier transform or descriptive Statistics applied.

For example, in perimetry periodically illuminating light marks of defined intensity and sufficient high frequency (over about 4Hz) used to make the visual system work to check. For the function test the electroencephalogram (EEG) is recorded and the stimulus response regarding the visual stimulus the amplitudes and the phase are analyzed. The stimulus response phase is one of the decisive diagnostic parameters in functional diagnostics.

The previous procedure is disadvantageous that the statistical uncertainty of the detection or the inaccuracy of the measurement is very high. The uncertainty and the inaccuracy result from the signal theory as a consequence of and related to length of the analysis window. The theory says that as the length of the Analysis window the statistical uncertainty and thus the inaccuracy increase, which is also sufficient in practical signal analysis proven and known. For a statistically better result would first have to increase the window length. From physiology, however, it is known that the phase is relative change quickly can and these changes are also diagnostically relevant. With a long analysis window passes the valuable information the phase change lost and the statistical uncertainty of the measurement result increases as a result of the changes not necessarily from.

The invention is based on the object to provide a method and an arrangement with which it is possible a causal phase response in periodic biosignals with a across from conventional processes better reliability and higher Detect speed with reduced computing effort and to eat.

According to the invention, the solution of Task in that periodic biosignals according to their physical and physiological origins are recorded that a condition observer parallel to the analyzed biological system it is established that the initial parameters of the biological system and of the observer can be evaluated using a Kalman filter and used to determine the phase.

In the method according to the invention the phase of a periodic biosignal is determined and for functional diagnostic Purposes. For example, an extended phase is compared to the healthy test objects an important indication of functional ones Problems of the examined biological system.

In the arrangement according to the invention is parallel to the investigated biological system, which is associated with a Condition model is modeled, a condition observer arranged, which, according to the system model, the state variable phase based on the Kalman filter estimates.

The advantage here is that the estimate the phase can take place continuously and no sliding or sequentially applied analysis window is necessary. So that will the analysis of the phase changes over time is only possible. in the Contrary to the relatively complicated theoretical background of this phase estimator the practical implementation is simple. Compared to conventional ones Procedure needed they use significantly less computing power, so a phase estimate in Real time possible is.

The invention is illustrated below the theoretical derivation and an embodiment explained in more detail. In the associated The drawings show:

1 - Block diagram of the observer concept

2 - system model

3 - Representation of the status observer for measuring the phase in periodic biosignals

4 - Course of the estimated phase for a harmonic of the frequency 8Hz and phase 2rad for the static Kalman factors 2 and 20

5 - Course of the estimated phase for a noisy harmonic of the frequency 8HZ and Pha se 2rad with an SNR from OdB (below) and dynamic Kalman factor (above)

6 - phase estimation of the signal as in 5 with static Kalman factors

7 - Results of the phase estimation (right) on real signals (left)

8th - additive superimposition of a harmonic of a frequency of 8 Hz with the noise and deliberate detuning of the analysis frequency (top) and the phase curve with rise (bottom)

The biological system that produces a periodic biosignal or responds to a periodic input signal is in 1 represented as a state model "real system". State equations (1) and (2) describe this system (bold letters in large letters stand for matrices, small ones for vectors): ẋ (t) = A * x (t) + B * u (t); x (0) = x (t 0 ) (1) y (t) = C x (t). (2)

For further considerations, we assume an additive signal model that sums up a harmonic oscillation and white, normally distributed noise: y (t) = ŷ · sin (ωt + φ (t)) + r P (t) (3)

The goal is a system model to construct, whose variable x (t) is the phase φ (t) of the signal to be examined y (t) represents. The phase cannot be measured directly because it is an argument is trigonometric function. Therefore, an auxiliary construction needed. Such a construction is a state observer that runs parallel to the examined system is arranged. The observer appreciates that State variable by minimizing an error function, which are the outputs of the real system and the observer. In this way after the error minimization has been completed, the state variable Phase can be measured directly.

The block diagram of the observer concept is in 1 shown. Since x (t) cannot be measured directly, x _m (t) is estimated in the observer. The inner loop in the observer minimizes the error of y _m (t) with respect to y (t) with the aid of the correction matrix K. The equations of state (4) and (5) then result for the observer. ẋ M (t) = A · x M (t) + B · u (t) + K · [y (t) - Y M (t)]., (4) y M (t) = Cx M (T). (5)

From (4) and (5) it follows: ẋ M (t) = (A - KC) x M (t) + B · u (t) + K · y (t). (6)

It is assumed that both systems have different initial conditions. This results in the observation error: e (t) = x (t) - x M (T). (7)

The observation error disappears iteratively with the aid of the correction matrix K, so that e (t) = 0 for t → ∞. (8th)

The dynamics and the stability of the estimation can be described with the differential equation of the observation error (9): ė (t) = ẋ (t) - ẋ M (T). (9)

Through conversion and further intermediate steps we get: ė (t) = (A - K · C) · e (t). (10)

According to the signal model (3), it can be expected that the signal under investigation is disturbed by noise. A Kalman filter is used to reduce the influence of noise. Taking the noise into account, the system is described by the following equations of state: ẋ (t) = A · x (t) + B · u (t) + M · r s (t) (11) System output: y (t) = Cx (t) + r P (t) (12) Observer : ẋ M (t) = A · x M (t) + B · u (t) + K (t) · [y (t) - C · x M (t)], (13) in which
K (t) is the correction matrix which has to achieve that
e (t) = x (t) - x _M (t) → 0,
e (t) is the observation error,
r _s (t) is the system noise, and
r _p (t) is the process noise.

To simplify the derivation, it is assumed that the noise components are broadband Gaussian zero mean processes with known covariances:

The noise components are independent of each other, that is

For a consistent estimate of x (t), the error performance must be minimized using the matrix K (t):

Assuming the stochastic relationships with regard to the covariance, a suitable correction matrix K (t) is derived in accordance with the Kalman filter: K (t) = eov e (t) · C T · R 1 p (T). (17)

The formula for the error covariance cov _ε (t) can be derived from the Kalman filter:

Estimation of the phase:

The signal under investigation is modeled according to (3) from the sum of a harmonic and noise: y (t) = y nl (t) + r P (t) = ŷ · sin (ωt + φ (t)) + r P (t), ω = 2πf. (19)

The phase results from the differential equation: φ (t) = –a · φ (t) + r s (t) a> 0. (20)

This results in the system model in 2 , The phase cannot be measured directly. An observer is therefore parallel to the system, in which direct access to the estimated phase φ _M (t) is possible. However, the non-linear part y _nl (t) in (19) is unfavorable for the observer concept. A suitable linearization y _l (φ (t), t) is required to establish a linear relationship between the state variable φ _M (t) and the To produce output y _M (t) according to (5). Based on Taylor linearization, the observer can be formulated as follows: M (t) = –a · φ M (t) + K (t) * (y (t) - y M (t)) (21) y M (t) = y l (φ M (t), t). (22)

According to (21) and (22), the observer is modeled as in 2 shown. As a result of the linearization at the operating point φ _B , (22) is linked with (5). This gives the factor C, which is used in (17) to determine the correction factor K (t): K (t) = cov e (t) · ŷ · cos (ωt + φ B (t)) · R 1 p (T). (23)

The phase to be determined is selected as the operating point φ Β (t) = φ M (t), (24) and the resulting differential equation for the phase

According to (25) the phase estimator can be modeled as in 3 shown.

The error covariance must be calculated for the phase estimation in y (t). From (18) it follows:

Equation (26) provides a simple solution if higher-frequency components are not taken into account in the error covariance. Based on (27) cos 2 (ωt + φ M (t)) = 1/2 + cos (2ωt + 2φ M (t)) (27) can (26) be simplified to:

With a suitable choice of parameter a in (28), high-frequency components as a result of temporal integration are suppressed, i.e. there is a low-pass behavior. Taking the low pass into account, (25) can be simplified:

The observer is represented in 3 , simplified. The system proposed in (29) can be used in particular for the phase estimation of harmonics in noise.

In 4 the course of the estimated phase for a harmonic of frequency 8Hz and phase 2rad for different Kalman factors is shown. The Kalman factors are static and are 2 or 20. As can be seen in the graph, the higher the Kalman factor, the slower the estimate. Static Kalman factors must be used where the time of the phase change is not known.

In 5 shows the curve (lower part) of the estimated phase for a noisy harmonic of frequency 8Hz and phase 2rad with an SNR (signal-to-noise ratio) of OdB and dynamic Kalman factor (upper part). If the time of the phase change is known, the Kalman factor can be constructed so that the change is followed first and the variance of the estimate is then minimized. For comparison is in 6 the phase estimate of the same signal as in 5 represented with static Kalman factors.

In 7 the results of the phase estimation (right column of the graphic) are shown on real signals. Light pulse sequences with a repetition rate of 8 pulses per second were used to stimulate the visual system, alternating between a pause (time from 0 to 2 of the time courses of the signals top left and bottom left) followed by light stimulation (time 2 to 5). The course of the EEG (Elektroen zephalogram, left column of the graphic) of two occipital positions after 16-fold stimulus-related averaging. Both time courses show a clear jump in the phase after the start of light stimulation.

The phase estimation becomes problematic with very noisy signals. In general, it is true that the phase is more robust against disturbances than the amplitudes, as is finally known in information technology. However, the question of the existence - i.e. the detection - of a causal phase must first be clarified in this border area, only then would the phase be estimated. In 8th is a harmonic of the frequency 8Hz additively superimposed on the noise starting at t = 4s, whereby the SNR is –10dB (upper curve in the graphic). The amplitude of the harmonic cannot be detected in the time domain. If the phase estimator is used with a specific detuning, here with a frequency of 7.8Hz, i.e. 0.2Hz less than the frequency of the harmonics, there is an increase of 1.2rad / s in the case of a causal phase (lower graph). This increase can be used directly in combination with a discriminator to detect the signal.

a

System parameters, selectable in the observer model

A, B

C., R

matrices in the state model of a system

φ (t)

phase in the system model, to be estimated size

φ _M (t)

phase in the observer model, measurable size

r _p (t)

process noise

r _s (t)

system noise

u (t)

input variable of a system in the state model

x (t)

state variable of a system in the state model

x _M (t) -

state variable of the observer in the state model

y (t)

output variable of a system in the state model

y _M (t)

output variable of the observer in the state model

y _l (φ (t), t)

Linearization operator for phase

Claims (2)

Method for the detection and measurement of the phase of periodic biosignals, characterized in that a) the signal to be analyzed, the phase of which is to be determined, is fed to a multiplier whose factor can be constant or variable over time, b) the product obtained is another Multiplier is supplied, the factor of which is a trigonometric function, the argument of which results from the product of the examined frequency with time, added to the measured phase, the frequency of the trigonometric analysis function corresponding to the frequency at which the phase is to be determined, or deviates from this frequency by a known amount, c) the measured phase is fed to a third multiplier, in which it is multiplied by a factor, d) the product obtained under b) has a positive input and the product obtained under c) the negative input of a difference generator can be supplied, e) that under d) he held difference is fed to a time integrator, at the output of which the measured phase to be determined is present. Arrangement for detection and measurement of the phase of periodic biosignals, characterized in that a) a condition observer parallel to the examined biological system is ordered b) the condition observer after the functions Claim 1 realized.